Influence of different disorder types on Aharonov-Bohm caging in the diamond chain

The linear diamond chain with fine-tuned effective magnetic flux has a completely flat energy spectrum and compactly localized eigenmodes, forming an Aharonov-Bohm cage. We study numerically how this localization is affected by different types of disorder (static and time-evolving) relevant to recent realizations of Aharonov-Bohm cages in periodically modulated optical waveguide arrays. We demonstrate robustness of localization under static and time-periodic disorder. In contrast, nonquenched (time-dependent) disorder leads to wave-packet spreading and delocalization.

Figure 1

(a) Schematic of optical waveguide array with effective magnetic flux implemented via periodic modulation of the waveguide depths (shown in red and blue) combined with transverse acceleration of the waveguide positions. (b) Band structure of the array in the Aharonov-Bohm (AB) cage limit $\mathrm{\Gamma }=\pi$. (c,d) Schematic of the compactly localized flat band modes: (c) Geometry-induced ${\beta }_{\text{FB}}=0$ flat band, and (d) ${\beta }_{\text{FB}}=\pm 2$ flat bands emerging due to the AB caging.

Figure 2

The (a) eigenstate density spectrum $\rho$ and (b) corresponding imbalance $\eta$ for diamond chain with $\mathrm{\Gamma }=\pi$ (AB cage limit) in the presence of on-site static disorder of strength $W=1$. (c,d) The corresponding quantities in the presence of static hopping disorder are plotted.

Figure 3

(a,b) Dynamics of the disorder-averaged participation ratio $\langle \text{PR}\rangle$ and second moment $\langle m_2\rangle$ obtained for excitations of the (a) ${\beta }_{\text{FB}}=0$ and (b) ${\beta }_{\text{FB}}=2$ CLS. (c,d) Disorder-averaged intensity profiles $\langle I_n\rangle$ at $z={10}^4$ for (c) ${\beta }_{\text{FB}}=0$ and (d) ${\beta }_{\text{FB}}=2$ CLS excitations. Averages are obtained over 50 disorder realizations.

Figure 4

Disorder-averaged $\langle \text{PR}\rangle$ (upper plot) and $\langle m_2\rangle$ (lower plot) after propagation length $z={10}^3$ for different stationary QD realizations vs. disorder strength $W$. (a) CLS from central FB (${\beta }_{\text{FB}}=0$) and (b) CLS from the periphery FB are initially excited in the lattice (from ${\beta }_{\text{FB}}=2$). Error bars illustrate the standard deviations of corresponding quantities.

Figure 5

Dynamics of the disorder-averaged participation ratio $\langle \text{PR}\rangle$ and second moment $\langle m_2\rangle$ obtained for excitations of the (a) ${\beta }_{\text{FB}}=0$ and (b) ${\beta }_{\text{FB}}=2$ CLS. (c,d) Intensity profiles $\langle I_n\rangle$ at $z=10000$ for (a) ${\beta }_{\text{FB}}=0$ and (b) ${\beta }_{\text{FB}}=2$ CLS excitations. Averages are obtained over 50 realizations of on-site disorders.

Figure 6

The averaged value of $\langle \text{PR}\rangle$ (upper plot) and $\langle m_2\rangle$ (lower plot) over the propagation length $z=1000$ for different stationary QD realizations vs. disorder strength $W$. (a) CLS from central FB (${\beta }_{\text{FB}}=0$) and (b) CLS from the periphery FB (${\beta }_{\text{FB}}=2$) are initially excited in the lattice.

Figure 7

Dynamics of the disorder-averaged participation ratio $\langle \text{PR}\rangle$ and second moment $\langle m_2\rangle$ obtained for excitations of the (a) ${\beta }_{\text{FB}}=0$ and (b) ${\beta }_{\text{FB}}=2$ CLS. (c,d) Intensity profiles $\langle I_n\rangle$ at (c) $z=10000$ for ${\beta }_{\text{FB}}=0$ and (d) ${\beta }_{\text{FB}}=2$ CLS excitations. Averages are obtained over 50 realizations of hopping disorders.

Figure 8

The on-site NQD power spectra obtained after averaging over ten different realizations of the set of random numbers for (a) resonant NQD and (b) off-resonant NQD.

Figure 9

Averaged values of participation ratio $\langle \text{PR}\rangle$ and second moment $\langle m_2\rangle$ over 50 propagation windows, which are obtained for CLS from (a) ${\beta }_{\text{FB}}=0$ excitation and (b) the CLS from ${\beta }_{\text{FB}}=2$ excitation and different types of NQDs are shown in log-scale. The mentioned propagation windows are obtained by dividing the total propagation length into 50 parts of equal length.